Fitting a linear model and something being truly linear in itself are two different things. A purely linear process will best described by mean coefficients in a linear equation and corresponding noise terms, with all residuals equally and independently distributed over all levels of measurement.  There are many ways for data to fail or to come close to failing these conditions.

r2 is often called the "coefficient of determination" and/or "proportion explained," but both of these phrases leave out the important premise that the meaning of this proportion is always conditional on the expectation that your data is best construed as a homogeneously distributed sum of squared differences from a population mean. Fitting a linear model basically means tiling your empirical distribution so that all data points are best approximated by the corners of a series of squares, the opposite corner of which is exactly the mean.

If you have nonlinear data, it is entirely possible and even likely to get a strong correlation coefficient for reasons more to do with chance than with the underlying process.

If you are working with quasilinear data you have a few popular options:

1) shrug and just fit the linear model, because that's the easiest for everyone to think about

2) fit a higher-order polynomial to pick up the nonlinear curvature--this is especially good to do if you have theoretical interest in the interpretation of the linear term. Higher order polynomials are all linear-modeling methods of decomposing nonlinear trajectories--important to distinguish between the empirical trajectory and the sort of model you are projecting on the empirical trajectory.

3) try a spline fit if you have a good algorithm or theoretical motivation for where to split the trajectory for separate linear regressions.

Best of luck,

Damian